260 
PROFESSOR KARL PEARSON, MATHEIRATICAL 
of Ancestral Heredity. It seems therefore worth while to prove the following 
proposition : 
Proposition I. — Any character not itself directly measurable, hut a function of 
physically measurahle characters and organs inherited according to the Law of 
Ancestral Hered.ity, ivdl itself he inherited according to that laiv. 
Thus if we assume intellectual and emotional characters to be ultimately a result 
j 
of physical conformation, we may be fairly certain that although we know neither 
the organs of which they are a function, nor the nature of that function, still they 
will be inherited according to the same law as that which holds for physically 
measurable oro-ans. 
Let y be the character in a parent, and let it be an unknown function f of the 
unknown physical organs x^, Xo, Xg, . . . x,,^, or let : 
y = f {x„ X., Xg, . . . xf .(i.). 
Let Ay denote the deviation from the mean value of the cha,racter y in some special 
individual, and A* the deviation from the mean of any x organ in the same individual, 
j'hen if these deviations be small compared with the mean values of the organs 
considered, we have from (i.) above : 
Ay = a^AXi + cuAx., + a^AXg + (in), 
where a^, a., . . . are constants independent of the individual variations. 
Let cr denote a standard deviation, p a coefficient of interorganic correlation, S a 
summation with regard to all individuals witli character y dealt with, and let them 
be n in number. Then : 
ncr; 
or 
= S {Ayf = S («! AXi + a-i Ax-y + cig AXg + ...)' 
= n {a\a\ -}- cdcrl^ + aga\ + , . . + + 2«ia3cr,-^o-.i-,Oc,.r3 + • • •) ^ 
a} = .(hi.), 
where S denotes a summation through the group of m organs. 
Let y' denote the character in an individual who is the offspring of the individual 
of character y, and x\, x'o, x'g . . . the corresponding organs. Then, if we do not 
suppose the nature of the function f to have changed in a single generation, we 
have : 
y'= f {x\, x' 2 , x'g . . . x\,f 
and 
Ay' = Ax'I + tto Ax' 2 + «3 A.r's +. 
= ^(fho-y,) + 2S .(v.). 
Let r be a coefficient of direct heredity expressing the correlation between parent 
